@amiloradovsky@functional.cafe Sure it behaves like a multiplication, if we forget what multiplication was, then take the common denominator of our 3 favourite operations (that kinda sorta resembled multiplication in some cases) and call that multiplication, cause we're feeling entitles. I guess you could call it "general multiplication", but that would very uncreative (and would also require forgetting the natural language you are using). I think this is quite unfortunate, and confusing in practice, even when doing algebra on paper.

A special diagonal matrix solves that one thing, but feels like a hack to me, or a high level API for people who really like matrices. I much rather my matrix be a vector of vectors (which indeed is not just any vector, unless you want to look at it that way) and have access to all fundamental operations on them, including the ones one would use to optimize a special higher level matrix/tensor type you present to mathematicians, for whatever perverted things they are into these days.